PROOF 56

Prove.

a² + b² = c²

On the Euclidean plane.

What you may use

A1one parallel through an external point
Euclid's fifth postulate (Playfair form)
A2angle sum of a triangle = 180°
Proof 49 — follows from A1
A3a square of side s exists
existence of squares — follows from A1
A4opposite sides of a parallelogram are equal
property of parallelograms — follows from A1
A5two sides and the included angle ⟹ congruent
SAS congruence condition — corresponding sides and angles are equal
A6area is the sum of non-overlapping pieces
additivity of area
A7area of a square = (side)²
convention for area
A8legs a, b ⟹ area = ab/2
area of a right triangle
A9a straight angle is 180°
angles on a line
A10angle measures > 0
positivity of angles
A11two angles respectively equal ⟹ similar
AA similarity condition — follows from A1
A12similar ⟹ equal corresponding side ratios
definition of similarity
A13one perpendicular from an external point
existence and uniqueness of the perpendicular
A14∠A, ∠B acute ⟹ H between A, B
location of the foot of the perpendicular
A15P–X–Q ⟹ PX + XQ = PQ
addition of segments
A16a/b = c/d ⟹ ad = bc
properties of proportions (b, d ≠ 0)
A17(a+b)² = a²+2ab+b²
expansion formula (Proof 23)
A18a(b+c) = ab + ac
distributive law
A19a+b = b+a, ab = ba
commutativity of addition and multiplication
A20(a+b)+c = a+(b+c)
associativity of addition
A21a+c = b+c ⟹ a = b
cancellation for addition
A22side lengths of a triangle > 0
positivity of length
Ea = b ⟹ same operation on both sides
properties of equality

Nothing outside this list may be cited.

Definition D1

In triangle ABC, let ∠C = 90° and set a = BC, b = CA, c = AB. ∠A, ∠B denote the interior angles at those vertices.

Proof.

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